Essentially, what you seem to be doing here is to write an extended version of a paper that previously appeared in a different venue (here, ArXiV). In this case, it is customary to include all of the relevant definitions, theorem, lemmas, and the like in the extended version.

The reason is that the longer paper - in a sense - replaces the shorter version. Thus, you would include them all. Nevertheless, you must state in your longer paper that it is an extended version of an earlier version of the paper (i.e., the short) paper.

Note that you can just cite parts of your ArXiV paper without including all of its results. It is probably not wise to pursue this approach, though. An ArXiV paper is unreviewed, so the reviewers of your journal paper are more likely to question the correctness of the theorem in it. Since also most publishers do not consider ArXiV papers to be a prior publication, there is no copyright or dual publication problem with just including theorems and proofs of your shorter paper.

Essentially, what you seem to be doing here is to write an extended version of a paper that previously appeared in a different venue (here, arXiv). In this case, it is customary to include all of the relevant definitions, theorem, lemmas, and the like in the extended version.

The reason is that the longer paper - in a sense - replaces the shorter version. Thus, you would include them all. Nevertheless, you must state in your longer paper that it is an extended version of an earlier version of the paper (i.e., the short paper).

Note that you can just cite parts of your arXiv paper without including all of its results. It is probably not wise to pursue this approach, though. An arXiv paper is unreviewed, so the reviewers of your journal paper are more likely to question the correctness of the theorem in it. Since also most publishers do not consider arXiv papers to be a prior publication, there is no copyright or dual publication problem with just including theorems and proofs of your shorter paper.